Question

in the figure below, b is between a and c, and c is between b and d. if bd=13, ac=11, and ad=18, find bc. bc =

Explanation:

Step1: Find AB + BC + CD

We know that \( AD = AB + BC + CD = 18 \) (since A to D is the total length). Also, \( BD = BC + CD = 13 \) (since B to D is BC + CD). And \( AC = AB + BC = 11 \) (since A to C is AB + BC).

Step2: Add AC and BD

If we add \( AC \) and \( BD \), we get \( AC + BD=(AB + BC)+(BC + CD)=AB + 2BC + CD \). We know \( AC = 11 \) and \( BD = 13 \), so \( AC + BD=11 + 13 = 24 \).

Step3: Solve for BC

We also know that \( AB + BC + CD=18 \). Let's denote \( AB + BC + CD = 18 \) as equation (1) and \( AB + 2BC + CD = 24 \) as equation (2). Subtract equation (1) from equation (2): \((AB + 2BC + CD)-(AB + BC + CD)=24 - 18\). Simplifying the left side gives \( BC \), and the right side gives \( 6 \). So \( BC = 6 \).

Answer:

\( 6 \)